Atwood's Machine Lab:


Purpose:
The purpose of the lab is to study acceleration in a pulley system with respect to varying masses. This can be used to analyze the relationship masses m1 and m2 on the Atwood's machine and the resulting accelerations.

Background:
In a pulley system, the larger mass will accelerate downward while the lighter one accelerates upward. The accelerations occur at the same rate. Also, the overall acceleration depends on the difference between m1 and m2 and the total mass of the system.

Hypothesis:

As the difference between m1 and m2 increases, the acceleration of the system will increase. Also, as total mass increases acceleration should decrease because it is more difficult to initiate motion of heavier objects.

Apparatus:

• Vernier computer interface (1)
• Computer with Logger Pro (1)
• Vernier Photogate with Ultra Pulley (1)
• mass sets (2)
• string (1)
• Ringstand (1)
• Clamp (1)
• Small cups (2)

Procedure:

Part 1 -

1. Set up the Atwood's machine, at least 40cm above floor
2. Connect the Photogate with Super Pulley to DIG/SONIC 1 of the interface
3. Open the file "10 Atwoods Machine" in the provided Logger Pro program
4. Place m1 of 200g on one side of the pulley and m2 of 200g on the other side
5. Position m1 as high as possible and click "collect" on Logger Pro
6. Let m1 drop
7. Record the data and acceleration
8. Move 5g from m2 to m1 and record the data.
9. Click "Examine" in Logger Pro for the region of increasing velocity on the newly generated graph. Then, click "Linear Fit" to find the slope which is also the acceleration. Record the acceleration.
10. Continue to move 5g increments of mass from m2 to m1 and repeat steps 6-7 for each combination.
11. Repeat this process five times.

Part 2 -

1. Repeat steps 1-3 of procedure part 1
2. Place 120 on m1 and 100 on m2.
3. Repeat steps 6-7 of procedure part 1
4. Add 20g increments to both sides of the pulley, keeping the difference 20g.
5. Record the masses and repeat steps 6-7 from Part 1.
6. Repeat this process five times

Data:

Table 1: Part 1 – Total Mass Constant
Trial	m1 (g)	m1 (g)	Acceleration (m/s2)	Δm (g)	mT (g)
1	202	200	0	2	402
2	207	195	0.1771	12	402
3	212	190	0.3811	22	402
4	217	184	0.6281	32	401
5	222	180	0.8297	42	402

Table 2: Part 2 – The Mass Difference Constant
Trial	m1 (g)	m1 (g)	Acceleration (m/s2)	Δm (g)	mT (g)
1	120	100	0.6538	20	220
2	140	120	0.5922	20	260
3	160	140	0.4111	20	300
4	180	160	0.3829	20	340
5	200	180	0.3487	20	380

Analysis:

3.)

LVang_-_Lab_10_-_Part_1_Acceleration_vs._Mass_Difference.png

The acceleration vs. mass difference graph shows a clear linear relationship. This means that the change in acceleration is directly proportional to the difference in mass between m1 and m2.
4.)

LVang_-_Lab_10_-_Part_2_Acceleration_vs._Total_Mass_(Linear).png

In the linear regression for the graph of acceleration vs. total mass, the acceleration decreases as the total weight within the system increases with the regression equation of y = -0.002x + 1.092 where (y) is acceleration and (x) is total mass.

LVang_-_Lab_10_-_Part_2_Acceleration_vs._Total_Mass.png

However, the residual plot for the linear regression of the Part 2 graph shows that a linear fit does not best represent the data. The results are more parabolic in nature. According to the graph, it seems a quartic regression best represents the trend of the relationship between acceleration and total mass in the Atwood's machine.

5.)
If a = 0.021(mass difference) - 0.061 and a = -0.002(total mass) + 1.092, then when combined,
2a = 0.021(mass difference) - 0.002(total mass) + 1.031.
When simplified, a = 0.0105(mass difference) - 0.001(total mass) + 0.5155.

If instead, the quartic equation where y = -7E-09x4 + 9E-06x3 - 0.003x2 + 0.759x - 54.27 is used, the combined equations will be
2a = 0.021(mass difference) - 7E-09(total mass)4 + 9E-06(total mass)3 - 0.003(total mass)2 + 0.759(total mass) - 54.331.


When this is simplified,
a = 0.0105(mass difference) - 3.5E-09(total mass)4 + 5E-06(total mass)3 - 0.0015(total mass)2 + 0.3795(total mass) - 27.1655.

Conclusion:

The hypothesis of this lab is supported by the data collected. It was determined that acceleration increases as the difference between m1 and m2 increases. This is probably due to the fact that F=ma. Therefore, as mass increase, net force increases in the direction of m1, causing a greater acceleration. Also, it was found that acceleration decreases as the total mass of the system increases. As the total mass became greater, the acceleration decreased in a non-linear fashion.